The meaning of "linear"

[dcuccia]

Hi!

I'm new to ASR, but thought the folks here might be able to help me understand what various groups mean when they say an amp or speaker is "linear."

I'm a (biomedical) engineer, and so when I hear the term, I understand it to mean that a system's output is *linearly* proportional to the power input; and a corollary - that it's frequency response is *independent* of the power input.

But I think I've heard many, knowledgeable, people call a system linear if it had a *flat* frequency response. Linear as in, it looks like a horizontal line?

Which brings me to the second part of my question - what *is* the term used for talking about linearly from an EE "signals and systems" perspective? It would seem pretty interesting to me to learn what amps and speakers have a similar frequency response or noise characteristic over a range of input power. E.g. how do Class A/B amp topologies typically perform, compared to Class D? Usually, the data I've seen (with a few caveats) for SNR have been for a low/reference power level, like 1W.


Thanks for humoring me!
David

 

[-Matt-]

Hi David,
Welcome to ASR.

I'm not an audio expert but I'll make an attempt...

You may see several different aspects of audio systems described as "linear" and the meaning can differ slightly depending on context.

For an amplifier when we talk about the linearity of its output it is easier to think of voltages rather than power. The job of an amplifier is to increase the magnitude of an input signal by a factor (this could be a fixed value like in a power amp or variable via the volume control on an integrated amp). To keep the sums simple let's say we are talking about a power amp with a gain factor of 10. Let's say that the input to the amp is a balanced xlr and can go up to 4V. If we put in a sequence of 1kHz test tones with RMS amplitudes of 0.5V, 1V, 1.5V, 2V we should expect to get outputs with RMS amplitudes of 5V, 10V, 15V, 20V. If we plotted this on a graph (output vs input) you would see a straight diagonal line with the gradient being the gain. The output would be "linear".

If an amplifier's response is nonlinear (in which case it will distort the input signal) then instead of a linear fit we would need to include higher order terms, quadratic, cubic, etc, to fully describe the relationship between input and output signals. The effect of these higher order terms will be to cause harmonics of any test tones played. In Amir's test results he shows the Fourier transform of all amplifier outputs and you can clearly see the peaks corresponding to the harmonics (although for a good amp these are at a vastly lower level than the main test tone frequency).

The other characteristic that you mentioned was the frequency response. Ideally we want the amplifier to have a "flat" frequency response (I guess you could call this "linear" but that wouldn't be the usual term). As you say, this would be a horizontal line if you plot output amplitude vs frequency. If an amplifier has a flat frequency response and you input a test signal with 1V RMS amplitude that sweeps across all audible frequencies then the RMS amplitude at the output should ideally stay at 10V throughout the frequency sweep (assuming the same gain value of 10 that we used before). If the amp displayed a larger gain for signals with higher frequency then the response would not be "flat" and it would therefore also distort the input signal.

Hopefully that goes some way to answering your question!

 

[NTK]

Hi!

I'm new to ASR, but thought the folks here might be able to help me understand what various groups mean when they say an amp or speaker is "linear."

I'm a (biomedical) engineer, and so when I hear the term, I understand it to mean that a system's output is *linearly* proportional to the power input; and a corollary - that it's frequency response is *independent* of the power input.

But I think I've heard many, knowledgeable, people call a system linear if it had a *flat* frequency response. Linear as in, it looks like a horizontal line?

Which brings me to the second part of my question - what *is* the term used for talking about linearly from an EE "signals and systems" perspective? It would seem pretty interesting to me to learn what amps and speakers have a similar frequency response or noise characteristic over a range of input power. E.g. how do Class A/B amp topologies typically perform, compared to Class D? Usually, the data I've seen (with a few caveats) for SNR have been for a low/reference power level, like 1W.


Thanks for humoring me!
David

A system is linear if it satisfy the superposition principle.

Linear system - Wikipedia

en.wikipedia.org

Flat frequency response is not a necessary characteristic of a linear system.

But of course the usage of the word does not necessarily has to be confined to the mathematical definition. Engineers and mathematicians do not own the word linear.

 

[boxerfan88]

To me, "linear" means at different loudness levels, the “system” frequency response keeps its shape/curve.
Quite easy to check with REW, taking a few measurement sweeps at 5dB intervals.

 

[Doodski]

For amplifiers... Linearity is the ability of a amp to maintain the signal integrity from the input of the amp to the output and... it also means the amplifier output wattage doubles for each halving of the load resistance.

 

[DVDdoug]

Be aware that "audiophiles" commonly use a LOT of vague-meaningless terminology.

Generally it means low distortion (perfectly linear would be zero distortion). Audio amplifiers are linear unless they are over-driven into clipping.

Dynamic compression and limiting are non-linear effects. Low-level signals pass-through the compressor linearly. Above the compression threshold the gain is reduced and the peaks are "pushed down" from they would otherwise be. Limiting is a kind of fast-compression and clipping is a "bad kind" of limiting.

A class-D amplifier might be described as non-linear because internally it's non-linear. But as a "black box" a class-D audio amplifier is a linear device.

Power supplies are often categorized as "switching" or "linear". A switching (AKA "switchmode") power supply is similar to a class-D amplifier and they are both are generally highly efficient. They both use variable-width pulses which are filtered-smoothed for constant DC out of a power supply, or "smooth-continuous AC" out of a power amplifier.

 

[wwenze]

There are two usages for the term "linear"

One is the mathematical definition, where something/some relationship, like a transfer function, follows a, well, straight line a.k.a. linear curve. Also refers to devices/components that exhibit this relationship e.g. resistors are a linear device because V=RI (or wtev formulae), as well as operating regions of complex circuits where this relationship is exhibited e.g. linear region of an amplifier.

The other, probably more often observed usage of the term "linear", is a circuit that is classically-designed, usually of a complexity level understoodable by the layman, which may or may not have anything to do with the mathematical term. Common offender: "Linear power supply" - It should only be called linear power supply if it includes a linear regulator, however people still call those 60Hz transformer power supplies with no regulation as "linear power supply".

 

[Doodski]

Common offender: "Linear power supply" - It should only be called linear power supply if it includes a linear regulator, however people still call those 60Hz transformer power supplies with no regulation as "linear power supply".

Thanks for mentioning this. I am always a bit confused as per why a non regulated power supply can be called a linear power supply. I went with the flow and joined in with other people using the term because it is so common but you are right it is not linear.

 

[Zapper]

A class-D amplifier might be described as non-linear because internally it's non-linear. But as a "black box" a class-D audio amplifier is a linear device.

True, but NO amplifier is linear if you look inside. BJTs, MOSFETs, vacuum tubes - they're all intrinsically nonlinear. It takes a bunch of resistively mediated negative feedback (local and global), linearizing circuits like current mirrors, and small signal perturbations to make the final result almost linear.

 

[IAtaman]

Linearity is a mathematical term. I am sure you are familiar with it. In EE, it is used in connection with its linear function sense. Simply put, a function would be called linear if it meets two criteria:
- if I take two values, add them up and apply the function to the added value, the output should be the same as applying the function separately to these values and adding the the results up later. f(a+b)=f(a)+f(b)
- if I take a value, multiply it by a scalar and apply the function , it should be the same as applying the function and multiplying it by the same scalar a.f(x)=f(a.x)

In control systems, control devices will take an input, change it some way or anther and spit out an output. That transformation of the input to output is called the transfer function. If the transfer function of a device is linear given certain limitations, than that device is said to be linear within those limitations.

No electrical system, (and probably no physical system?) is fully linear; there is always some form of noise and/or distortion that gets added to the output that is not contained in the input. And there are always limitations, boundaries - no system is truly linear for all input and output values. But, as long as the non-linearities can be quantified, measured and accounted for, and are small enough for our purposes, we can label a system linear, meaning "linear enough". I suspect it would have a similar meaning in your field as well as the underlying math would be the same. But it is also well understood that non-engineering fields do like to take liberties with the meaning of well defined terms, so who knows

A flat frequency response would indicate linearity because the output level of the device under question is not affected by the frequency and is only related to the input signal level. I understand it might be confusing as we are used to seeing inputs and outputs mapped to a graph. But in this case the input is constant and is not shown on the graph, and only the output is shown as a function of frequency and flat means output is also constant like the input.

There is not a single *term* for linearity. A system's linearity can be affected by many things; frequency, output power, impedance of the load etc. so all of these needs to be looked at. People do ask for a single, simple and easy to understand metric to *gauge* performance like you do quite understandibly. ASR uses best case SINAD as a general indication of that (and gets a lot of heat for it.)

 

[MaxwellsEq]

But I think I've heard many, knowledgeable, people call a system linear if it had a *flat* frequency response. Linear as in, it looks like a horizontal line?

A "flat frequency response" is definitely not a correct use of the term. I might consider a flat frequency response from 0Hz to infinity Hz as a potential indicator of a linear system (i.e. no matter what frequency you input, the gain G is constant).
In HiFi, small signal amplifiers are the easiest to consider. Amplifiers can be DC-coupled, which in theory means that they will have some gain down to 0Hz, but loudspeakers definitely do not enjoy lots of DC. Meanwhile at the higher frequencies most amplifiers roll off above the audio band, often for stability reasons. So most small signal audio amplifiers are "passband" devices covering 10Hz to, say 100kHz.
But, just because a small signal amplifier has a frequency response + or - 0.1dB between 20 and 20kHz is no evidence that it is a linear device.
A small signal amplifier could deliberately compress, meaning G is not constant for all input signals. It could have a very high noise floor, which means that G becomes non-predictable. It could change the signal and distort it.

All of the above is only about relatively predictable devices. Transducers, such as speakers and cartridges are not even able to have a flat frequency response.

Which brings me to the second part of my question - what *is* the term used for talking about linearly from an EE "signals and systems" perspective? It would seem pretty interesting to me to learn what amps and speakers have a similar frequency response or noise characteristic over a range of input power. E.g. how do Class A/B amp topologies typically perform, compared to Class D? Usually, the data I've seen (with a few caveats) for SNR have been for a low/reference power level, like 1W.

I don't think "linear" should ever be used in the context of a HiFi system.

To answer your question, for many measurements, there is little difference between Class A/B and Class D. Neither are linear - they may have flat frequency responses in their passband (20Hz to 20kHz), but they do have noisefloors and they do distort. Class A/B are simpler (for the layman to understand). Class D are more complex above the audible frequency band. They interact with speakers in a slightly different way.

 

[IAtaman]

I might consider a flat frequency response from 0Hz to infinity Hz as a potential indicator of a linear system

That statement is useless from a practical and engineering point of view; it is purely academic; it does not offer an explanation to the question asked at best, in fact probably creates more confusion without any context . He asked what people mean when they say linear for FR graphs, he did not ask for applied mathematics class. So I wonder, what exactly is your point?

 

[MaxwellsEq]

That statement is useless from a practical and engineering point of view; it is purely academic; it does not offer an explanation to the question asked at best, in fact probably creates more confusion without any context . He asked what people mean when they say linear for FR graphs, he did not ask for applied mathematics class. So I wonder, what exactly is your point?

Please calm down.

My point: you can never use the term "linear" in its proper mathematical/engineering sense to describe any audio recording and playback system. You especially can not use it (as the OP asked) solely when only considering frequency response because a linear frequency response would require constant gain (or some other function) at all frequencies.

So if anyone says "my amp is linear because it is + or - 0.01dB between 10Hz and 50kHz", they are formally incorrect.

 

[solderdude]

Linear Audio

 

[IAtaman]

Please calm down.

My point: you can never use the term "linear" in its proper mathematical/engineering sense to describe any audio recording and playback system. You especially can not use it (as the OP asked) solely when only considering frequency response because a linear frequency response would require constant gain (or some other function) at all frequencies.

So if anyone says "my amp is linear because it is + or - 0.01dB between 10Hz and 50kHz", they are formally incorrect.

I am not saying what you said was incorrect, I am saying it is irrelevant and confusing for the person who is asking the question. Linear in the sense you are talking about is a mathemetical concept. It is not from our world. So in my view saying that there is no equivalent of mathematical linearity in our universe and leaving it there is not helpful at all, assuming your intention was to help OP to understand better what people mean when they say linear.

Sorry if it came across aggressive. I got irritated because I perceived your answer to be academic flexing at the expense of the OP. I am not saying that is what you were doing. But that is my perception and also my problem, not yours. Apologies.

 

[MaxwellsEq]

Linear Audio

There's a lot of it about

 

[MaxwellsEq]

I am not saying what you said was incorrect, I am saying it is irrelevant and confusing for the person who is asking the question. Linear in the sense you are talking about is a mathemetical concept. It is not from our world. So in my view saying that there is no equivalent of mathematical linearity in our universe and leaving it there is not helpful at all, assuming your intention was to help OP to understand better what people mean when they say linear.

Sorry if it came across aggressive. I got irritated because I perceived your answer to be academic flexing at the expense of the OP. I am not saying that is what you were doing. But that is my perception and also my problem, not yours. Apologies.

OK understood, and thank you for apologising! I'm actually a practical engineer who has spent hours under mixing consoles dealing with spilt wine. I also have a degree in electronics, and 50% of my degree was mathematics. These days, with software simulations, you don't need the mathematics, but when SIN and COSINE were the best functions on your calculator, you really needed to understand mathematics to do electronics.

I was just answering the question "The meaning of linear". But you are correct, in the audio world it's a degraded term, meaning whatever the person using it wants it to mean. So not very helpful when encountered.

 

[-Matt-]

It is probably also worth mentioning this plot that Amir includes in his reviews...

This is showing the differences relative to an ideal "linear" response as a function of input level. ("Linear" as in output=input*gain).

You can see that amplifiers typically do very well for larger signals (right of plot) but can struggle for very small signals (left of plot) because noise can become similar in scale to the signal.

 

[MaxwellsEq]

It is probably also worth mentioning this plot that Amir includes in his reviews...

This is showing the differences relative to an ideal "linear" response. ("Linear" as in output=input*gain).

You can see that amplifiers typically do very well for larger signals (right of plot) but can struggle for very small signals (left of plot) because noise can become similar in scale to the signal.

In my opinion, this is a better use of "linearity" than the frequency response domain. I think it says something about a consistent implementation of gain from below the noise floor to the maximum.

Some of the confusion in the use of the meaning of "linear" perhaps comes from the meaning "a straight line". In this case, the device should have straight line. In the frequency response graph, it should be a straight line in the audio passband. People talk about a "straight wire with gain".

 

[IAtaman]

in the audio world it's a degraded term, meaning whatever the person using it wants it to mean. So not very helpful when encountered.

You are probably right, it is used a bit too liberally in the audio world, like every other technical term. But it does have a meaningful engineering sense as well, in that if the non-linearity errors are known, measured and small enough to have no impact on the performance of the system in its designed operating range parameters, than that system can be labeled as linear. As always the case in audio, fight usually stems from what is small enough to have an impact term.

It is probably also worth mentioning this plot that Amir includes in his reviews...

I think this is a good example of another important aspect of linearity, but also might be confusing in a similar way to the frequency graph, in that it does not map input to output directly, it compares input level to the relative output level, expecting the ideal case to be 0. Makes it easier to see linearity errors, but requires a bit effort from people who are not used to this types of graphs.